Chapter 12 Statistics 2012 Pearson Education, Inc.

Chapter 12: Statistics 12.1 Visual Displays of Data 12.2 Measures of Central Tendency 12.3 Measures of Dispersion 12.4 Measures of Position 12.5 The Normal Distribution 2012 Pearson Education, Inc.

Measures of Dispersion Section 12-3 Measures of Dispersion 2012 Pearson Education, Inc.

Measures of Dispersion Range Standard Deviation Interpreting Measures of Dispersion Coefficient of Variation 2012 Pearson Education, Inc.

Measures of Dispersion Sometimes we want to look at a measure of dispersion, or spread, of data. Two of the most common measures of dispersion are the range and the standard deviation. 2012 Pearson Education, Inc.

Range For any set of data, the range of the set is given by Range = (greatest value in set) – (least value in set). 2012 Pearson Education, Inc.

Example: Range of Sets Solution The two sets below have the same mean and median (7). Find the range of each set. Set A 1 2 7 12 13 Set B 5 6 8 9 Solution Range of Set A: 13 – 1 = 12. Range of Set B: 9 – 5 = 4. 2012 Pearson Education, Inc.

Standard Deviation One of the most useful measures of dispersion, the standard deviation, is based on deviations from the mean of the data. 2012 Pearson Education, Inc.

Example: Deviations from the Mean Find the deviations from the mean for all data values of the sample 1, 2, 8, 11, 13. Solution The mean is 7. Subtract to find deviation. Data Value 1 2 8 11 13 Deviation –6 –5 4 6 13 – 7 = 6 The sum of the deviations for a set is always 0. 2012 Pearson Education, Inc.

Standard Deviation The variance is found by summing the squares of the deviations and dividing that sum by n – 1 (since it is a sample instead of a population). The square root of the variance gives a kind of average of the deviations from the mean, which is called a sample standard deviation. It is denoted by the letter s. (The standard deviation of a population is denoted the lowercase Greek letter sigma.) 2012 Pearson Education, Inc.

Calculation of Standard Deviation Let a sample of n numbers x1, x2,…xn have mean Then the sample standard deviation, s, of the numbers is given by 2012 Pearson Education, Inc.

Calculation of Standard Deviation The individual steps involved in this calculation are as follows Step 1 Calculate the mean of the numbers. Step 2 Find the deviations from the mean. Step 3 Square each deviation. Step 4 Sum the squared deviations. Step 5 Divide the sum in Step 4 by n – 1. Step 6 Take the square root of the quotient in Step 5. 2012 Pearson Education, Inc.

Example Solution Find the standard deviation of the sample 1, 2, 8, 11, 13. Solution The mean is 7. Data Value 1 2 8 11 13 Deviation –6 –5 4 6 (Deviation)2 36 25 16 Sum = 36 + 25 + 1 + 16 + 36 = 114 2012 Pearson Education, Inc.

Example Solution (continued) Divide by n – 1 with n = 5: Take the square root: 2012 Pearson Education, Inc.

Interpreting Measures of Dispersion A main use of dispersion is to compare the amounts of spread in two (or more) data sets. A common technique in inferential statistics is to draw comparisons between populations by analyzing samples that come from those populations. 2012 Pearson Education, Inc.

Example: Interpreting Measures Two companies, A and B, sell small packs of sugar for coffee. The mean and standard deviation for samples from each company are given below. Which company consistently provides more sugar in their packs? Which company fills its packs more consistently? Company A Company B 2012 Pearson Education, Inc.

Example: Interpreting Measures Solution We infer that Company A most likely provides more sugar than Company B (greater mean). We also infer that Company B is more consistent than Company A (smaller standard deviation). 2012 Pearson Education, Inc.

Chebyshev’s Theorem For any set of numbers, regardless of how they are distributed, the fraction of them that lie within k standard deviations of their mean (where k > 1) is at least 2012 Pearson Education, Inc.

Example: Chebyshev’s Theorem What is the minimum percentage of the items in a data set which lie within 3 standard deviations of the mean? Solution With k = 3, we calculate 2012 Pearson Education, Inc.

Coefficient of Variation The coefficient of variation is not strictly a measure of dispersion, it combines central tendency and dispersion. It expresses the standard deviation as a percentage of the mean. 2012 Pearson Education, Inc.

Coefficient of Variation For any set of data, the coefficient of variation is given by for a sample or for a population. 2012 Pearson Education, Inc.

Example: Comparing Samples Compare the dispersions in the two samples A and B. A: 12, 13, 16, 18, 18, 20 B: 125, 131, 144, 158, 168, 193 Solution The values on the next slide are computed using a calculator and the formula on the previous slide. 2012 Pearson Education, Inc.

Example: Comparing Samples Solution (continued) Sample A Sample B Sample B has a larger dispersion than sample A, but sample A has the larger relative dispersion (coefficient of variation). 2012 Pearson Education, Inc.